891 research outputs found

    TheoryGuru: A Mathematica Package to Apply Quantifier Elimination Technology to Economics

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    We consider the use of Quantifier Elimination (QE) technology for automated reasoning in economics. There is a great body of work considering QE applications in science and engineering but we demonstrate here that it also has use in the social sciences. We explain how many suggested theorems in economics could either be proven, or even have their hypotheses shown to be inconsistent, automatically via QE. However, economists who this technology could benefit are usually unfamiliar with QE, and the use of mathematical software generally. This motivated the development of a Mathematica Package TheoryGuru, whose purpose is to lower the costs of applying QE to economics. We describe the package's functionality and give examples of its use.Comment: To appear in Proc ICMS 201

    Non-linear Real Arithmetic Benchmarks derived from Automated Reasoning in Economics

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    We consider problems originating in economics that may be solved automatically using mathematical software. We present and make freely available a new benchmark set of such problems. The problems have been shown to fall within the framework of non-linear real arithmetic, and so are in theory soluble via Quantifier Elimination (QE) technology as usually implemented in computer algebra systems. Further, they all can be phrased in prenex normal form with only existential quantifiers and so are also admissible to those Satisfiability Module Theory (SMT) solvers that support the QF_NRA. There is a great body of work considering QE and SMT application in science and engineering, but we demonstrate here that there is potential for this technology also in the social sciences.Comment: To appear in Proc. SC-Square 2018. Dataset described is hosted by Zenodo at: https://doi.org/10.5281/zenodo.1226892 . arXiv admin note: substantial text overlap with arXiv:1804.1003

    Bridging symbolic computation and economics: a dynamic and interactive tool to analyze the price elasticity of supply

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    It is not possible to achieve the objectives and skills of a program in economics, at the secondary and undergraduate levels, without resorting to graphic illustrations. In this way, the use of educational software has been increasingly recognized as a useful tool to promote students' motivation to deal with, and understand, new economic concepts. Current digital technology allows students to work with a large number and variety of graphics in an interactive way, complementing the theoretical results and the so often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. Thanks to the symbolic and numerical capabilities of Mathematica, these applications allow the user to interact with the graphical and analytical information in real time. However, Mathematica is a commercially distributed application which makes it difficult for teachers and students to access. The main goal of this paper is to present a new dynamic and interactive tool, created with Mathematica and available in the Computable Document Format. This format allows anyone with a computer to use, at no cost, the PES(Linear)-Tool, even without an active Wolfram Mathematica license. The PES(Linear)-Tool can be used as an active learning tool to promote better student activity and engagement in the learning process, among students enrolled in socio-economic programs. This tool is very intuitive to use which makes it suitable for less experienced users.Funding Agency Portuguese Foundation for Science and Technology UID/ECO/04007/2019info:eu-repo/semantics/publishedVersio

    Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets

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    In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other is a modal fixpoint language with expressions for optimality, rationality and belief. Fixpoints are used to form expressions for common belief and for iterated elimination of non-optimal strategies.Comment: 16 pages, Proc. 11th International Workshop on Computational Logic in Multi-Agent Systems (CLIMA XI). To appea

    SC-Square: Overview to 2021.

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    Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition

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    There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. The present work extends to have ML select the variable ordering directly, and to try a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human constructed heuristics for the problem: Brown's heuristic and sotd. On this dataset all of the ML approaches outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201

    Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds

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    Given a formula in quantifier-free Presburger arithmetic, if it has a satisfying solution, there is one whose size, measured in bits, is polynomially bounded in the size of the formula. In this paper, we consider a special class of quantifier-free Presburger formulas in which most linear constraints are difference (separation) constraints, and the non-difference constraints are sparse. This class has been observed to commonly occur in software verification. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of non-difference constraints, in addition to traditional measures of formula size. In particular, we show that the number of bits needed per integer variable is linear in the number of non-difference constraints and logarithmic in the number and size of non-zero coefficients in them, but is otherwise independent of the total number of linear constraints in the formula. The derived bound can be used in a decision procedure based on instantiating integer variables over a finite domain and translating the input quantifier-free Presburger formula to an equi-satisfiable Boolean formula, which is then checked using a Boolean satisfiability solver. In addition to our main theoretical result, we discuss several optimizations for deriving tighter bounds in practice. Empirical evidence indicates that our decision procedure can greatly outperform other decision procedures.Comment: 26 page
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